The human immune deficiency virus (HIV) harms the body by attacking an immune cell called the CD4 cell and, thus, making the body more vulnerable to other illness-causing germs. Because of this, CD4 cell count per milliliter of blood ([1]} [1]}) can be used to track the progression of HIV. A person with untreated HIV will experience a series of stages: acute HIV infection (2-6 weeks), stage one (1-5 years), stage two (6-9 years), stage three (9-11 years), which is also called the acquired immunodeficiency syndrome (AIDS), and stage four (11-12 years). In the acute HIV infection, the CD4 cell counts go down from 1200 to 900; in stage one, the CD4 cell counts drop slowly from 900 to 500; during stage two, it may go down from 500 to 350 ([3]} [3]}). This CD4 cell counts data (available in the refund package in R) was analysed by [5]} ([5]}) to construct the corrected confidence interval for sparse functional data. It includes observed CD4 cell counts for 366 infected individuals from \(-18\)  to 42 months (Figure REF ), where 0 represents seroconversion, the moment when the antibody becomes detectable in the blood. The observation period in this data set includes acute HIV infection and stage one, or stage two for some patients who are in worse situations.
